I have the problem 2e^x = squareroot(10e^x) +3. the solution must be in exact form only.

Let sqrt(2e^x) = y

y^2 = 2 e^x

sqrt(10e^x) = sqrt5*sqrt(2e^x)
= sqrt5*y

y^2 - sqrt5*y -3 = 0, a quadratic

B^2 - 4AC = 17

y = [sqrt5 +/- sqrt17]/2

Knowing y, use the natural log function to solve for x in exact closed form. It will probably look messy.